Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Factorization wikipedia , lookup
History of algebra wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Elementary algebra wikipedia , lookup
Field (mathematics) wikipedia , lookup
System of polynomial equations wikipedia , lookup
Homomorphism wikipedia , lookup
Module 1 Lesson 2 A Literal Equation is an equation with two or more variables. • You can "rewrite" a literal equation to isolate any one of the variables using inverse operations. • When you rewrite literal equations, you may have to divide by a variable or variable expression. Step 1 Locate the variable you are asked to solve for in the equation. Step 2 Identify the operations on this variable and the order in which they are applied. Step 3 Use inverse operations to undo operations and isolate the variable. A. Solve x + y = 15 for x. x + y = 15 Since y is added to x, subtract y –y –y from both sides to undo the x = –y + 15 addition. B. Solve pq = x for q. pq = _x_ p p Since q is multiplied by p, divide both sides by p to undo the multiplication. Solve 5 – b = 2t for t. 5 – b = 2t Locate t in the equation. Since t is multiplied by 2, divide both sides by 2 to undo the multiplication. Solve for the indicated variable. 1. for h 2. P = R – C for C 3. 2x + 7y = 14 for y 4. for m 5. for C 1. H = 3V A 2. y = 14 – 2x 7 3. C = R – P 4. m = x(k – 6 ) 5. C = Rt + S The formula C = d gives the circumference of a circle C in terms of diameter d. The circumference of a bowl is 18 inches. What is the bowl's diameter? Leave the symbol in your answer. Locate d in the equation. Since d is multiplied by , divide both sides by to undo the multiplication. Now use this formula and the information given in the problem. The formula C = d gives the circumference of a circle C in terms of diameter d. The circumference of a bowl is 18 inches. What is the bowl's diameter? Leave the symbol in your answer. Now use this formula and the information given in the problem. The bowl's diameter is inches. Every Real Number is either rational or irrational. We refer to these sets as subsets of the real numbers, meaning that all elements in each subset are also elements in the set of real numbers. Numbers Examples Natural Numbers Whole Numbers 2,3,4,17 0,2,3,4,17 Integers -5,-2,0,2,5 Rational Numbers Irrational Numbers 1 5 1 2 , ,.4 ,0,.6 2 1 5 3 2, , 3 25 is a rational number because 25 5. Example Consider the following set of numbers. 1 3, 0, , .95, , 8, 2 List the numbers in the set that are: a. Natural Numbers b. Whole Numbers c. Integers d. Rational Numbers e. Irrational Numbers f. Real numbers 16 Example Consider the following set of numbers. 1 3, 0, , .95, , 8, 2 16 List the numbers in the set that are: a. Natural Numbers: √16 = 4, so that is the only Natural Number b. Whole Numbers: 0 , √16 c. Integers: -3, 0, √16 d. Rational Numbers: -3, 0, ½ , .95, √16 e. Irrational Numbers: √8 f. Real numbers: All of the numbers listed above! Property Addition Multiplication a+b=b+a ab = ba (a + b) + c = a + (b + c) (ab)c = a(bc) Identity a+0=a a*1=a Inverse a + (-a) = 0 a * 1/a = 1 Commutative Associative Distributive a(b + c) = ab + ac It doesn’t matter how you swap addition or multiplication around…the answer will be the same! Rules: Samples: Commutative Property of Addition Commutative Property of Addition a+b = b+a 1+2 = 2+1 Commutative Property of Multiplication Commutative Property of Multiplication ab = ba (2x3) = (3x2) It doesn’t matter how you group (associate) addition or multiplication…the answer will be the same! Rules: Samples: Associative Property of Addition Associative Property of Addition (a+b)+c = a+(b+c) (1+2)+3 = 1+(2+3) Associative Property of Multiplication (ab)c = a(bc) Associative Property of Multiplication (2x3)4 = 2(3x4) What can you add to a number & get the same number back? ZERO What can you multiply a number by and get the number back? ONE Rules: Identity Property of Addition a+0 = a Identity Property of Multiplication a(1) = a Samples: Identity Property of Addition 3+0=3 Identity Property of Multiplication 2(1)=2 Think opposites! The Inverse property uses the inverse operation to get to the identity! Rules: Inverse Property of Addition a+(-a) = 0 Samples: Inverse Property of Addition 3+(-3)=0 Inverse Property of Multiplication a(1/a) = 1 Inverse Property of Multiplication 2(1/2)=1 You can distribute the coefficient through the parenthesis with multiplication and remove the parenthesis. Rule: a(b+c) = ab+bc Samples: 4(3+2)=4(3)+4(2)=12+8=20 • 2(x+3) = 2x + 6 • -(3+x) = -3 - x 1. 2. 3. 4. 5. 6. 7. 8. 9. 6+8=8+6 5 + (2 + 8) = (5 + 2) + 8 12 + 0 =12 5(2 + 9) = (5 2) + (5 9) 4 (8 2) = (4 8) 2 5/9 9/5 = 1 5 24 = 24 5 18 + -18 = 0 -34 1 = -34 1. 2. 3. 4. 5. 6. 7. 8. 9. Commutative Associative Identity Distributive Associative Inverse Commutative Inverse Identity All real numbers have closure. The Closure property states the if a and b are real numbers then: • a + b is a real number • ab is a real number. So, if you add two rational numbers, your sum will be rational. Also, if you add two irrational numbers, that sum will be irrational.